Latest content added for UNT Digital Library Partner: UNT Librarieshttp://digital.library.unt.edu/explore/partners/UNT/browse/?sort=added_d&fq=str_degree_department:Department+of+Mathematics&fq=untl_collection:UNTETD2014-04-23T20:20:45-05:00UNT LibrariesThis is a custom feed for browsing UNT Digital Library Partner: UNT LibrariesA Comparative Study of Non Linear Conjugate Gradient Methods2014-04-23T20:20:45-05:00http://digital.library.unt.edu/ark:/67531/metadc283864/<p><a href="/ark:/67531/metadc283864/"><img alt="A Comparative Study of Non Linear Conjugate Gradient Methods" title="A Comparative Study of Non Linear Conjugate Gradient Methods" src="/ark:/67531/metadc283864/thumbnail/"/></a></p><p>We study the development of nonlinear conjugate gradient methods, Fletcher Reeves (FR) and Polak Ribiere (PR). FR extends the linear conjugate gradient method to nonlinear functions by incorporating two changes, for the step length αk a line search is performed and replacing the residual, rk (rk=b-Axk) by the gradient of the nonlinear objective function. The PR method is equivalent to FR method for exact line searches and when the underlying quadratic function is strongly convex. The PR method is basically a variant of FR and primarily differs from it in the choice of the parameter βk. On applying the nonlinear Rosenbrock function to the MATLAB code for the FR and the PR algorithms we observe that the performance of PR method (k=29) is far better than the FR method (k=42). But, we observe that when the MATLAB codes are applied to general nonlinear functions, specifically functions whose minimum is a large negative number not close to zero and the iterates too are large values far off from zero the PR algorithm does not perform well. This problem with the PR method persists even if we run the PR algorithm for more iterations or with an initial guess closer to the actual minimum. To improve the PR algorithm we suggest finding a better weighing parameter βk, using better line search method and/or using specific line search for certain functions and identifying specific restart criteria based on the function to be optimized.</p>Centers of Invariant Differential Operator Algebras for Jacobi Groups of Higher Rank2014-04-23T20:20:45-05:00http://digital.library.unt.edu/ark:/67531/metadc283833/<p><a href="/ark:/67531/metadc283833/"><img alt="Centers of Invariant Differential Operator Algebras for Jacobi Groups of Higher Rank" title="Centers of Invariant Differential Operator Algebras for Jacobi Groups of Higher Rank" src="/ark:/67531/metadc283833/thumbnail/"/></a></p><p>Let G be a Lie group acting on a homogeneous space G/K. The center of the universal enveloping algebra of the Lie algebra of G maps homomorphically into the center of the algebra of differential operators on G/K invariant under the action of G. In the case that G is a Jacobi Lie group of rank 2, we prove that this homomorphism is surjective and hence that the center of the invariant differential operator algebra is the image of the center of the universal enveloping algebra. This is an extension of work of Bringmann, Conley, and Richter in the rank 1case.</p>Natural Smooth Measures on the Leaves of the Unstable Manifold of Open Billiard Dynamical Systems2014-03-26T09:30:20-05:00http://digital.library.unt.edu/ark:/67531/metadc278917/<p><a href="/ark:/67531/metadc278917/"><img alt="Natural Smooth Measures on the Leaves of the Unstable Manifold of Open Billiard Dynamical Systems" title="Natural Smooth Measures on the Leaves of the Unstable Manifold of Open Billiard Dynamical Systems" src="/ark:/67531/metadc278917/thumbnail/"/></a></p><p>In this paper, we prove, for a certain class of open billiard dynamical systems, the existence of a family of smooth probability measures on the leaves of the dynamical system's unstable manifold. These measures describe the conditional asymptotic behavior of forward trajectories of the system. Furthermore, properties of these families are proven which are germane to the PYC programme for these systems. Strong sufficient conditions for the uniqueness of such families are given which depend upon geometric properties of the system's phase space. In particular,
these results hold for a fairly nonrestrictive class of triangular configurations of
scatterers.</p>Minimality of the Special Linear Groups2014-03-26T09:30:20-05:00http://digital.library.unt.edu/ark:/67531/metadc279280/<p><a href="/ark:/67531/metadc279280/"><img alt="Minimality of the Special Linear Groups" title="Minimality of the Special Linear Groups" src="/ark:/67531/metadc279280/thumbnail/"/></a></p><p>Let F denote the field of real numbers, complex numbers, or a finite algebraic extension of the p-adic field. We prove that the special linear group SLn(F) with the usual topology induced by F is a minimal topological group. This is accomplished by first proving the minimality of the upper triangular group in SLn(F). The proof for the upper triangular group uses an induction argument on a chain of upper triangular subgroups and relies on general results for locally compact topological groups, quotient groups, and subgroups. Minimality of SLn(F) is concluded by appealing to the associated Lie group decomposition as the product of a compact group and an upper triangular group. We also prove the universal minimality of homeomorphism groups of one dimensional manifolds, and we give a new simple proof of the universal minimality of S∞.</p>Properties of Bicentric Circles for Three-Sided Polygons2014-03-26T09:30:20-05:00http://digital.library.unt.edu/ark:/67531/metadc278727/<p><a href="/ark:/67531/metadc278727/"><img alt="Properties of Bicentric Circles for Three-Sided Polygons" title="Properties of Bicentric Circles for Three-Sided Polygons" src="/ark:/67531/metadc278727/thumbnail/"/></a></p><p>We define and construct bicentric circles with respect to three-sided polygons. Then using inherent properties of these circles, we explore both tangent properties, and areas generated from bicentric circles.</p>On Groups of Positive Type2014-03-24T20:07:29-05:00http://digital.library.unt.edu/ark:/67531/metadc277804/<p><a href="/ark:/67531/metadc277804/"><img alt="On Groups of Positive Type" title="On Groups of Positive Type" src="/ark:/67531/metadc277804/thumbnail/"/></a></p><p>We describe groups of positive type and prove that a group G is of positive type if and only if G admits a non-trivial partition. We completely classify groups of type 2, and present examples of other groups of positive type as well as groups of type zero.</p>Polish Spaces and Analytic Sets2014-03-24T20:07:29-05:00http://digital.library.unt.edu/ark:/67531/metadc277605/<p><a href="/ark:/67531/metadc277605/"><img alt="Polish Spaces and Analytic Sets" title="Polish Spaces and Analytic Sets" src="/ark:/67531/metadc277605/thumbnail/"/></a></p><p>A Polish space is a separable topological space that can be metrized by means
of a complete metric. A subset A of a Polish space X is analytic if there is a Polish
space Z and a continuous function f : Z —> X such that f(Z)= A. After proving that
each uncountable Polish space contains a non-Borel analytic subset we conclude that there exists a universally measurable non-Borel set.</p>Physical Motivation and Methods of Solution of Classical Partial Differential Equations2014-03-24T20:07:29-05:00http://digital.library.unt.edu/ark:/67531/metadc277898/<p><a href="/ark:/67531/metadc277898/"><img alt="Physical Motivation and Methods of Solution of Classical Partial Differential Equations" title="Physical Motivation and Methods of Solution of Classical Partial Differential Equations" src="/ark:/67531/metadc277898/thumbnail/"/></a></p><p>We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.</p>Multifractal Analysis of Parabolic Rational Maps2014-03-24T20:07:29-05:00http://digital.library.unt.edu/ark:/67531/metadc278398/<p><a href="/ark:/67531/metadc278398/"><img alt="Multifractal Analysis of Parabolic Rational Maps" title="Multifractal Analysis of Parabolic Rational Maps" src="/ark:/67531/metadc278398/thumbnail/"/></a></p><p>The investigation of the multifractal spectrum of the equilibrium measure for
a parabolic rational map with a Lipschitz continuous potential, φ, which satisfies
sup φ < P(φ)
x∈J(T)
is conducted. More specifically, the multifractal spectrum or spectrum of singularities, f(α) is studied.</p>Existence of Many Sign Changing Non Radial Solutions for Semilinear Elliptic Problems on Annular Domains2014-03-24T20:07:29-05:00http://digital.library.unt.edu/ark:/67531/metadc278251/<p><a href="/ark:/67531/metadc278251/"><img alt="Existence of Many Sign Changing Non Radial Solutions for Semilinear Elliptic Problems on Annular Domains" title="Existence of Many Sign Changing Non Radial Solutions for Semilinear Elliptic Problems on Annular Domains" src="/ark:/67531/metadc278251/thumbnail/"/></a></p><p>The aim of this work is the study of the existence and multiplicity of sign changing nonradial solutions to elliptic boundary value problems on annular domains.</p>